# Tagged Questions

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### Stochastic integration by parts to obtain Kailath Segall identity for iterated stochastic integrals?

If $(M_t)_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I_0=1$, $I_1(t)=M_t$ and for $n \geq 2$ $$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$ By noting ...
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### Reference request: Stochastic integration and martingale theory on the whole real line

I'm looking for a thorough treatment of stochastic integration and/or martingale theory on the whole real line, i.e. a way to construct a Brownian motion $(B_s)_{s \in \mathbb{R}}$ (if a two-sided BM ...
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### Probability that d-Brownian Motion ,d>3, avoids a set A

In other words, the probability that Brownian motion stays within $A^{c}$. So far I found that it is 1, for random cylinders and thorns (http://www.math.upenn.edu/~pemantle/papers/burdzy.pdf). What ...
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### weak convergence of the solutions to stochastic heat equation

$W(t,x)=\sum_ic_ie_i(x)B^i_t$ is a Brownian motion in $L^2(R^d)$, where $\{e_i\}$ is the standard orthogonal basis and $\sum_ic_i^2<\infty$. $$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$ ...
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### On the superior of generalized Ornstein-Uhlenbeck process

Let us consider a generalized O-U process $X_t \in L^2[0, 1]$ defined by the following spde: $dX_t = \frac{1}{2}\partial_x^2X_t + dW_t,$ $\partial_x X_t(0) = \partial_x X_t(1) = 0,$ $X_0 = 0,$ ...
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### Numerical computation of Skorokhod integral

How can I numerically compute the Skorokhod integral of a non-adapted process? If it is adapted, that is easy since the integral is just an Ito integral. I have found that computing the Malliavin ...
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### Inflated independent samples for Monte Carlo estimation

In my particular problem, running an MCMC is too expensive, so I'm looking for a simple MC estimator, which would partially inherit the correlated samples of MCMC, yet would not require computing ...
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### Strong Markov property for Poisson point process

The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail. Here is what I mean exactly. ...
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### Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation $$dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0,$$ where ...
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### a special filtration satisfying $0$-$1$ law

Let $\xi$ be a uniformly random variable on $[0,1]$ defined on some probability space $(\Omega,\mathcal{F})$. Define the process $\xi_t:=\min(\xi,t)$ for $0\le t\le 1$. And let ...
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### Can every discrete martingale be embedded in a continuous martingale?

Let $(X_k)_{k=0,1,..., n}$ be a discrete martingale defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$. I would like to know whether there exists a (continuous) martingale ...
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### The jump and the left martingale of semimartingale

Let $X_{t}$ be a semimartingale. Define $\Delta X_{t} = X_{t}- X_{t-}$. For fixed $s> 0$, $\Delta X_{s}$ and $X_{s-}$ are two random variable. Are they independent to each other? I think the ...
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### The regularity of Levy process

There is a property for continuous Markov process that each point $y$ in its state space is hit with positive probability one starting from any interior point $x$. This property is called the ...
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### The distribution of Jump gaps of Levy process

Assume $X_{t}$ is a Levy process with triplet $(\sigma^{2}, \lambda, \nu)$, here $\nu$ is the Levy measure of $X_{t}$. Define $\tau_{1},\tau_{2},\dots$ be the time gap between the successive jumps ...
I would like to know more about the two-dimensional processes derived from Brownian motion by the following stochastic differential equation (in the Ito sense) $$dX_t = f(X_t) dt + ... 0answers 54 views ### Ito formula for max(X,0) where X is a semimartingale Has anyone ever applied the Ito formula on |X^+|^2 for X^+ = \max(X,0) with X(t) = X(0) + M(t) + V(t), where M(t) is a local martingale and V(t) is bounded variation process. I found it in ... 1answer 121 views ### a dominated convergence theorem for martingale (II) The question is presented in a dominated convergence theorem for martingale Let \{(X_1^n, X_2^n)\}_n be a sequence of martingales defined some probability space. (which means ... 1answer 117 views ### Domino Shuffling and Warren's process In this paper by Nordenstam, it is shown that a certain interlacing particle process that arises from uniformly random Aztec diamond tilings is amazingly similar to Warren's process. One of the ... 1answer 93 views ### Can <.> of a martingale determine it only? Let \Omega be the space of continuous functions defined on [0,1]. Define the canonical process B by$$B_t(\omega)=\omega_t,~ \forall\omega\in\Omega$$Let us equip \Omega with the usual ... 1answer 117 views ### a L^1 convergence for backward martingale I have a question which may be naive, but I can not find the related result in the classical reference such as "Foundations of Modern Probability" and "Probability"(Billingsley). So if someone knows ... 2answers 144 views ### On the existence and uniqueness of solution to SPDE with nonlinear growth coefficients Consider the SPDE$$\frac{\partial}{\partial t}u_t(x) = \frac{\kappa}{2}\frac{\partial^2}{\partial x^2}u_t(x) + u_t(x)(K-u_t(x)) + \sigma u_t(x) \xi(t,x),$$where (t,x)\in {\mathbb R}_+\times ... 2answers 203 views ### Any suggestions on a rigorous stochastic differential equations book? I have been looking through some books and they are not very rigorous. Any suggestions would be great. 0answers 101 views ### On the infinitesimal generator of a 1-dimensional stochastic heat equation: core and explicit form Denote E = C([0, 1]). I am consider a 1-dimentional stochastic heat equation on h: \partial_tu(t, x) = \partial_x^2u(t, x) - V'(u(t, x)) + \dot{W}(t, x), for all (t, x) \in (0, ... 1answer 116 views ### On the expected value of a random integral: Is it possible to find the expected value of u(t) in terms of the following information:$$u(t)=\int_{0}^{t}(t-s)(f(s)+(T-s)Y)X_sds$$where: X_s is a wide sense stationary process with known ... 1answer 82 views ### Multiplicative version of Novikov inequality for Ito integral It is clear that Ito isometry E(∫^t_0fdW)^2=E(∫^t_0f^2dt) can be written in the multiplicative form as E(∫^t_0fdW\cdot∫^t_0gdW)=E(∫^t_0f⋅gdt). Is it possible to obtain the multiplicative version ... 3answers 301 views ### When is a continuous path stochastic process be representable as diffusion or Ito process? When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be? 0answers 63 views ### Time change for non-homogeneous Markov processes Background: Let C be the space of continuous function on [0,T], f, \sigma \in C bounded with \sigma^2 \geq \varepsilon > 0 and let X=(X_t)_{t\in [0,T]} be a diffusion process of ... 1answer 80 views ### construction of a approximate martingale everyone. Given a probabilistic space (\Omega, \mathcal{F}_t, \mathbb{P}) and a martingale (M_t)_{t\leq 1} on it. Suppose$$M_1\stackrel{\mathbb{P}}{\sim}\mu$$where \mu is a probability ... 1answer 174 views ### Stochastic integral with respect to discontinuous martingale in my research, I need to deal with a stochastic integral with respect to a compensated poisson process, namely,  \int_0^t f(X_t) dM_t, where M(t) = N(t) - \int_0^t \lambda(s)ds. The integrand ... 1answer 417 views ### On the pathwise uniqueness of solutions of SDEs(Stochastic Differential Equations) Suppose that (\Omega,\mathscr{F},P) is a complete probability space equipped a filtration \{\mathscr{F}_t\} satisfying the usual conditions. B_t is a 1-dimentional Brownian motion with respect ... 2answers 196 views ### Probability of winding number of 2D Brownian Motion Let B_t be a 2D Brownian Motion with B_0 = (1,0). Now, express B_t in polars, that is, B_t = (r(t), \theta(t)). Let \tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}. What is \mathbb{P}[\tau ... 1answer 119 views ### Example of Girsanov change of density with finite relative entropy, but with infinite integral over squared changed drift Let (\Omega, (\mathcal F_t), \mathbb P) denote the usual Wiener space where \Omega = C[0,\infty), etc., and where (W_t)_{t \geq 0} denotes the Wiener process. Let Z \in L^1(\mathbb P) with Z ... 0answers 191 views ### distribution of integral of exponential of wiener process I am absolute newbie to stochastic calculus and have to solve a weighted hazard rates integral, where the hazard rates are stochastic, their logarithm governed by arithmetic Ornstein-Uhlenbeck (OU) ... 0answers 108 views ### Is there a theory of SDEs whose coefficients are themselves adapted processes (i.e. “may depend on the past”)? Is there an existence and uniqueness theorem for SDEs of the following type: dW_{t}=d\tilde{W}_{t}+\mu\left(\left(W_{s}\right)_{0\le s\le t},t\right)dt, where \tilde{W}_{t} is say ... 1answer 236 views ### Markov Chain: state reduction Hi I am trying to understand a proof in a paper (written by Isaac Sonin), I don't know if anyone could give me a clarification on the following: Firstly we have a Markov chain \{Y_k\} with finite ... 1answer 247 views ### Upper bound on the maxima of ratio of expectation of quantities under Gaussian measure Let \lambda,\eta >0 be given, and u:\mathbb{R}\rightarrow \mathbb{R} be a real valued function. Define$$\Delta(u)= \frac{\int u(h) \exp(-\eta ...
I'm looking for a more general Feynman-Kac formula that works in the case of jump-diffusion processes. I know that, given a pure diffusion process like $$dS_t=\mu_tdt+\sigma_tdW_t,$$ if $u(t,s)$ ...